Found problems: 85335
2015 Israel National Olympiad, 7
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number.
[list=a]
[*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$.
[*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$.
[/list]
1975 Bulgaria National Olympiad, Problem 3
Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ($a_0\ne0$) such that $|f(x)|\le1$ for every $x\in[-1,1]$. Prove that
(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which
(b) $|a_0|\le4$.
[i]V. Petkov[/i]
2005 India IMO Training Camp, 1
Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties:
(i) $a \in M$ and $b \in M$;
(ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$.
Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$
Indonesia MO Shortlist - geometry, g7
In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$
2009 District Olympiad, 1
Find all non-negative real numbers $x, y, z$ satisfying $x^2y^2 + 1 = x^2 + xy$, $y^2z^2 + 1 = y^2 + yz$ and $z^2x^2 + 1 = z^2 + xz$.
1987 IberoAmerican, 1
Find the function $f(x)$ such that
\[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \]
for $x\not=0,x\not=1,x\not=-1$.
2022 South East Mathematical Olympiad, 7
Let $a,b$ be positive integers.Prove that there are no positive integers on the interval $\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg)$.
2025 Sharygin Geometry Olympiad, 8
The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. Points $K$ and $L$ lie on $AC$, $BD$ respectively in such a way that $CK=AP$ and $DL=BP$. Prove that the line joining the common points of circles $ALC$ and $BKD$ passes through the mass-center of $ABCD$.
Proposed by:V.Konyshev
Oliforum Contest V 2017, 4
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and
$$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$.
(Salvatore Tringali)
2016 Tuymaada Olympiad, 8
The flights map of air company $K_{r,r}$ presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed $2*m^r$ .
2010 Purple Comet Problems, 1
If $125 + n + 135 + 2n + 145 = 900,$ find $n.$
2006 MOP Homework, 3
For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$,
$$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$
2011 Argentina Team Selection Test, 4
Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.
2005 Bulgaria National Olympiad, 5
For positive integers $t,a,b,$a $(t,a,b)$-[i]game[/i] is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.
1998 India National Olympiad, 1
In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.
2009 Peru IMO TST, 1
Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$
2002 Estonia National Olympiad, 1
Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.
1967 Poland - Second Round, 4
Solve the equation in natural numbers $$
xy+yz+zx = xyz + 2.
$$
2023 239 Open Mathematical Olympiad, 7
The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$.
[i]Proposed by David Brodsky[/i]
2022 Malaysia IMONST 2, 3
Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.
1971 Miklós Schweitzer, 1
Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$.
[i]J. Erdos[/i]
1996 USAMO, 2
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
2020 LMT Fall, 5
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8?$
[i]Proposed by Alex Li[/i]
2002 Regional Competition For Advanced Students, 4
Let $a_0, a_1, ..., a_{2002}$ be real numbers.
a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies:
it is smaller or equal to $1/4$.
b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ?
c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ :
the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.
2018 Puerto Rico Team Selection Test, 5
In the square shown in the figure, find the value of $x$.
[img]https://cdn.artofproblemsolving.com/attachments/0/1/4659d5afa5b409d9264924735297d1188b0be3.png[/img]